Comment by aap_ | Hacker News Reader

Comment by aap_ | Hacker News Reader

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The article reads more like a trolling attempt. Geometric/Clifford algebra is incredibly useful and by throwing away its product you lose a lot of the power of the algebra. It's like saying matrix multiplication is not useful and you really want to be multiplying and adding numbers in various ways. After all GA/CA elements can always be mapped to elements of a matrix algebra. To get rid of the idea of linear transformations and that they should compose just doesn't sound well thought through.

I don't know what sort of crackpots he's talking about, personally i haven't heard of them, only the accusations. If the author can't separate the math from the people who developed and/or popularized it, too bad. Does GA magically give intuitive explanations for all sorts of weird things? no. Can you formulate a lot of stuff much more efficiently and concisely, and does it help gain new perspective on some things? yes, absolutely. It provides a wonderful framework for expressing geometric ideas.

> As for pure math—it seems like research mathematics readily talks about and uses Clifford Algebra, but is uninterested in or specificaly avoids the terms and concepts that are specific to Hestenes’ “Geometric Algebra”. I can speculate as to why: even by the 90s/00s, GA had gotten a bad reputation because of its tendency to attract bad mathematicians and full-on crackpots.

https://alexkritchevsky.com/2024/02/28/geometric-algebra.htm...

No example of such a crackpot is given. The author just claims this without evidence. I've heard this sort of argument before but it's not very clear what it refers to.

> Can you formulate a lot of stuff much more efficiently and concisely, and does it help gain new perspective on some things? yes, absolutely. It provides a wonderful framework for expressing geometric ideas.

Can you elaborate on what stuff does it help to formulate much more efficiently and concisely?

I think one of the coolest examples is probably classical mechanics. See the SIBGRAPI 2021 videos on https://bivector.net/doc.html

All of these stuff can be done in normal linear algebra. Some (not all) of the operations can be done more efficiently with GA in low dimensions. It is neither more concise nor more intuitive to understand than normal linear algebra.

There are ten thousand examples I want to give of why you're wrong. We have to start somewhere so here's a favourite, the "universal projection formula":

(A.B)/B

Projects any A onto any B, in any number of dimensions and with any signature (eg hyperbolic/Euclidean/elliptic). A and B can be lines, planes, points, and with a conformal or anti de Sitter metric a sphere or hyperboloid etc ("blades").

It works because A.B is dimension independently the object "orthogonal to A and containing B or vice versa". And division by B will intersect that orthogonal object with B.

Concise, intuitive, and powerful. What's the linear algebra formula you'd consider to be comparable?

> All of these stuff can be done in normal linear algebra

On its own that is not a very strong argument. What you can do in linear algebra can be done by scalar add multiply and divide. That additions can be done with logical gates does not mean that programming an accounting application with logical gates as primitives is a good idea.

> It is neither more concise nor more intuitive to understand than normal linear algebra.

The real contention is this one. I have met people who hold opposite views on this

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